Showing papers on "Trigonometric interpolation published in 1978"
TL;DR: In this article, the authors complete and extend the work of Kilgore [8] on the optimal nodes in polynomial interpolation in the Banach space with the usual norm lI,fi~ := max l.f'(x)l. nkxZ:h
75 citations
TL;DR: Nikolskii-type inequalities for trigonometric polynomials and pth power integrable functions, 0
Abstract: Nikolskii-type inequalities, thus inequalities between different metrics of a function, are established for trigonometric polynomials and pth power integrable functions, 0
60 citations
01 Jan 1978
TL;DR: In this article, a characterization of interpolation sequences in terms of a "uniform separation condition" is given, which is essentially the best possible condition of its kind, and an explicit example of such a condition is given.
Abstract: In [3] Korevaar and Dixon have considered an interpolation problem for entire functions (stemming from work by Pavlov [4]), which is connected with Muntz — approximation on arcs and Macintyre's conjecture. In this note we give a characterization of interpolation sequences in terms of a “uniform separation condition”. We also give an explicit example which shows that a condition for interpolation sequences in [3] is essentially best possible of its kind.
27 citations
TL;DR: For the random trigonometric polynomial N 2 s,(/) cosn0, n-l where g(t), 0 < / < 1, are dependent normal random variables with mean zero, variance one and joint density function (1.2) as discussed by the authors.
Abstract: For the random trigonometric polynomial N 2 s,(/)cosn0, n-l where g„(t), 0 < / < 1, are dependent normal random variables with mean zero, variance one and joint density function \\M\\U2(2v)-\"/2exp[ Ci/2)ä'Mä] where A/-1 is the moment matrix with py = p, 0 < p < 1, i ¥*j, i,j m 1, 2.N and a is the column vector, we estimate the probable number of zeros. 1. Consider the random trigonométrie polynomial N (1.1) (9) = $(t, 9) =% gn(t) cos n9, 71=1 where g„(t), 0 < t < 1, are dependent normal random variables with mean zero, variance one and joint density function (1.2) |Af|1/2(27r)~\"/2exp[(1/2)0' Ma], where M ~x is the moment matrix with pi} = p, 0 < p < 1, i =£j, i,j = 1, 2, . . . , N, and ä is the column vector whose transpose is 5' = (gx(t),..., gN(t)). In this paper we calculate the probable number of zeros of (1.1). We prove the following. Theorem 1. In the interval 0 < 9 < 2tt all save certain exceptional set of functions , where ex is any positive number less than 1/13. The particular case when p = 0, that is the case when gn(t) are independent normal random variables, was considered by Dunnage [3] and proved the following. Received by the editors May 26, 1975 and, in revised form, July 19, 1976. AMS (MOS) subject classifications (1970). Primary 60-XX. O American Mathematical Society 1978 57 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
27 citations
TL;DR: In this paper, the authors derived results on the interpolation of complete quasinormed operator ideals, mainly for the absolutelyp-summing and the s-number ideals defined by Pietsch, by estimating the K-functional of Peetre.
Abstract: We derive results on the interpolation of complete quasinormed operator ideals, mainly for the absolutelyp-summing and thes-number idealsSps defined by Pietsch By estimating theK-Functional of Peetre, we get that the interpolation ideal (Sp1s,Sp2s),θ,p is contained inSps and is even equal to it in the case of the approximation numbers A similar fact is proved for absolutely (p, q)-summing operators, interpolating the first index We show further that the absolutelyp-summing operators onc0 are contained in the complex interpolation space (πp1(co), πp2(co))[θ]
24 citations
01 Nov 1978
TL;DR: In this paper, the authors provide a formula for Kergin interpolation based on the Newton form for univariate polynomial interpolation and show that it converges for analytic functions of several variables.
Abstract: : Very little seems to be known about polynomial interpolation of multivariate functions. However, Kergin recently established the existence and uniqueness of a natural extension of univariate interpolation to a multivariate setting. In this paper we provide a formula for Kergin interpolation. This formula is based on the Newton form for univariate polynomial interpolation. The error in approximating by Kergin interpolation is also obtained in a convenient form which allows us to assess the quality of this scheme. In particular, we establish that Kergin interpolation converges for analytic functions of several variables.
11 citations
TL;DR: It is shown that substantial errors in interpolation by means of existing discrete Fourier transform techniques are generally present for finite sequences, even if the function sampled is band-limited, and a modified approach is proposed which provides significantly more accurate information.
Abstract: It is shown that substantial errors in interpolation by means of existing discrete Fourier transform (DFT) techniques are generally present for finite sequences, even if the function sampled is band-limited. A modified approach is proposed which provides significantly more accurate information.
10 citations
01 Jan 1978
10 citations
6 citations
TL;DR: In this paper, a nonnegative Fourier transform with energy spectrum G (j\omega) and a discrete Fourier series (DFS) is used to determine the energy spectrum of a function.
Abstract: The following form of the factorization problem is considered: Given a function g(t) vanishing for |t| > a , and with a nonnegative Fourier transform G(j\omega) , rind a function f(t) with energy spectrum G (j\omega) and such that f(t)=O outside the interval (0, a). A numerical method for determining f(t) is developed involving only the discrete Fourier series.
5 citations
TL;DR: In this paper, the authors proposed a method to compute the dynamic response of a skeletal space frame when subjected to earthquake ground motions using trigonometric interpolation and a simple recursive algorithm which is fast and accurate.